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gsBSplineBasis< T > Class Template Reference

Detailed Description

template<class T>
class gismo::gsBSplineBasis< T >

A univariate B-spline basis.

Template Parameters
Tcoefficient type
KnotVectorTypethe type of knot vector to use
+ Inheritance diagram for gsBSplineBasis< T >:
+ Collaboration diagram for gsBSplineBasis< T >:

Public Types

typedef gsBSplineTraits< 0, T >
::Basis 
BoundaryBasisType
 Associated Boundary basis type.
 
typedef gsBSplineTraits< 1, T >
::Geometry 
GeometryType
 Associated geometry type.
 
typedef Basis_t ** iterator
 Iterators on coordinate bases.
 
typedef T Scalar_t
 Coefficient type.
 

Public Member Functions

_activeLength () const
 Returns length of the ``active" part of the knot vector.
 
gsMatrix< index_tactive (const gsMatrix< T > &u) const
 Returns the indices of active (nonzero) functions at points u, as a list of indices. More...
 
void active_cwise (const gsMatrix< T > &u, gsVector< index_t, d > &low, gsVector< index_t, d > &upp) const
 
void active_into (const gsMatrix< T > &u, gsMatrix< index_t > &result) const
 Returns the indices of active basis functions at points u, as a list of indices, in result. A function is said to be active in a point if this point lies in the closure of the function's support. More...
 
gsMatrix< index_tallBoundary () const
 Returns the indices of the basis functions that are nonzero at the domain boundary.
 
void anchor_into (index_t i, gsMatrix< T > &result) const
 Returns the anchors (greville points) of the basis.
 
gsMatrix< T > anchors () const
 Returns the anchor points that represent the members of the basis. There is exactly one anchor point per basis function. More...
 
void anchors_into (gsMatrix< T > &result) const
 Returns the anchors (greville points) of the basis.
 
const gsBasis< T > & basis (const index_t k) const
 Helper which casts and returns the k-th piece of this function set as a gsBasis.
 
const_iterator begin () const
 
iterator begin ()
 
int borderKnotMult () const
 Returns the multiplicity of the first ``significant" knot (i.e., the m_p+1st). If it is different from the multiplicity of the corresponding knot at the end, returns zero.
 
gsMatrix< index_tboundary (boxSide const &s) const
 Returns the indices of the basis functions that are nonzero at the domain boundary as single-column-matrix.
 
BoundaryBasisType::uPtr boundaryBasis (boxSide const &s)
 Gives back the boundary basis at boxSide s.
 
gsMatrix< index_tboundaryOffset (boxSide const &s, index_t offset) const
 
bool check () const
 Check the BSplineBasis for consistency.
 
uPtr clone ()
 Clone methode. Produceds a deep copy inside a uPtr.
 
gsMatrix< index_tcoefSlice (short_t dir, index_t k) const
 Returns all the basis functions with tensor-numbering. More...
 
gsSparseMatrix< T > collocationMatrix (gsMatrix< T > const &u) const
 Computes the collocation matrix w.r.t. points u. More...
 
Self_t & component (short_t i)
 For a tensor product basis, return the 1-d basis for the i-th parameter component.
 
const Self_t & component (short_t i) const
 For a tensor product basis, return the (const) 1-d basis for the i-th parameter component.
 
virtual uPtr componentBasis (boxComponent b) const
 Returns the basis that corresponds to the component.
 
virtual uPtr componentBasis_withIndices (boxComponent b, gsMatrix< index_t > &indices, bool noBoundary=true) const
 Returns the basis that corresponds to the component. More...
 
virtual void compute (const gsMatrix< T > &in, gsFuncData< T > &out) const
 Computes function data. More...
 
void connectivity (const gsMatrix< T > &nodes, gsMesh< T > &mesh) const
 
virtual void connectivityAtAnchors (gsMesh< T > &mesh) const
 
virtual gsBasis::uPtr create () const
 Create an empty basis of the derived type and return a pointer to it.
 
short_t degree (short_t i) const
 Degree with respect to the i-th variable. If the basis is a tensor product of (piecewise) polynomial bases, then this function returns the polynomial degree of the i-th component.
 
void degreeDecrease (short_t const &i=1, short_t const dir=-1)
 Lower the degree of the basis by the given amount, preserving knots multiplicity.
 
void degreeElevate (short_t const &i=1, short_t const dir=-1)
 Elevate the degree of the basis and preserve the smoothness.
 
void degreeIncrease (short_t const &i=1, short_t const dir=-1)
 Elevate the degree of the basis by the given amount, preserve knots multiplicity.
 
void degreeReduce (short_t const &i=1, short_t const dir=-1)
 Reduce the degree of the basis by the given amount, preserve smoothness.
 
gsMatrix< T > deriv (const gsMatrix< T > &u) const
 Evaluate the derivatives,. More...
 
gsMatrix< T > deriv2 (const gsMatrix< T > &u) const
 Evaluates the second derivatives of active (i.e., non-zero) functions at points u. More...
 
void deriv2_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
 Evaluate the second derivatives of all active basis function at points u. More...
 
void deriv2Single_into (index_t i, const gsMatrix< T > &u, gsMatrix< T > &result) const
 Evaluate the (partial) derivatives of the i-th basis function at points u into result.
 
void deriv_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
 Evaluates the first partial derivatives of the nonzero basis function. More...
 
void derivSingle_into (index_t i, const gsMatrix< T > &u, gsMatrix< T > &result) const
 Evaluates the (partial) derivatives of the i-th basis function at points u into result. More...
 
std::string detail () const
 Return a string with detailed information on the basis.
 
gsDomain< T > * domain () const
 
short_t domainDim () const
 Dimension of the (source) domain. More...
 
domainEnd () const
 Returns the ending value of the domain of the basis.
 
domainStart () const
 Returns the starting value of the domain of the basis.
 
size_t elementIndex (const gsVector< T > &u) const
 Returns an index for the element which contains point u.
 
gsMatrix< T > elementInSupportOf (index_t j) const
 Returns (the coordinates of) an element in the support of basis function j.
 
void elementSupport_into (const index_t i, gsMatrix< index_t, 1, 2 > &result) const
 Returns span (element) indices of the beginning and end of the support of the i-th basis function.
 
void elevateContinuity (int const &i=1)
 Elevate spline continuity at interior knots by i.
 
const_iterator end () const
 
iterator end ()
 
void enforceOuterKnotsPeriodic ()
 Moves the knot vectors to enforce periodicity.
 
gsMatrix< T > eval (const gsMatrix< T > &u) const
 Evaluate the function,. More...
 
virtual void eval_into (const gsMatrix< T > &u, gsMatrix< T > &result) const
 Evaluates nonzero basis functions at point u into result. More...
 
virtual void eval_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
 Evaluate an element of the space given by coefs at points u.
 
std::vector< gsMatrix< T > > evalAllDers (const gsMatrix< T > &u, int n, bool sameElement=false) const
 Evaluate all derivatives upto order n,. More...
 
virtual void evalAllDers_into (const gsMatrix< T > &u, int n, std::vector< gsMatrix< T > > &result, bool sameElement=false) const
 Evaluate the nonzero functions and their derivatives up to order n at points u into result. More...
 
virtual void evalAllDersSingle_into (index_t i, const gsMatrix< T > &u, int n, gsMatrix< T > &result) const
 Evaluate the basis function i and its derivatives up to order n at points u into result.
 
virtual void evalDerSingle_into (index_t i, const gsMatrix< T > &u, int n, gsMatrix< T > &result) const
 Evaluate the (partial) derivative(s) of order n the i-th basis function at points u into result.
 
virtual void evalFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
 Evaluate the function described by coefs at points u, i.e., evaluates a linear combination of coefs x BasisFunctions, into result. More...
 
virtual void evalSingle_into (index_t i, const gsMatrix< T > &u, gsMatrix< T > &result) const
 Evaluate the i-th basis function at points u into result.
 
void expandCoefs (gsMatrix< T > &coefs) const
 Helper function for transforming periodic coefficients to full coefficients.
 
index_t firstActive (T u) const
 Returns the index of the first active (ie. non-zero) basis function at point u Takes into account non-clamped knots.
 
gsBasisFun< T > function (index_t i) const
 Returns the i-th basis function as a gsFunction. More...
 
void getComponentsForSide (boxSide const &s, std::vector< Basis_t * > &rr) const
 Returns the components for a basis on the face s.
 
virtual T getMaxCellLength () const
 Get the maximum mesh size, as expected for approximation error estimates.
 
virtual T getMinCellLength () const
 Get the minimum mesh size, as expected for inverse inequalities.
 
 gsBSplineBasis (KnotVectorType KV, const bool periodic=false)
 Construct BSpline basis of a knot vector.
 
 gsBSplineBasis (std::vector< KnotVectorType > KV)
 Compatibility constructor with input an std::vector containing a single knotvector.
 
 gsBSplineBasis (const T u0, const T u1, const unsigned interior, const int degree, const unsigned mult_interior=1, const bool periodic=false)
 Construct a BSpline basis. More...
 
unsigned index (unsigned i, unsigned j, unsigned k=0) const
 
index_t index (gsVector< index_t, d > const &v) const
 
bool indexOnBoundary (const gsVector< index_t, d > &ind) const
 Returns true iff the basis function with multi-index ind is on the boundary.
 
bool indexOnBoundary (const index_t m) const
 Returns true iff the basis function indexed m is on the boundary.
 
bool inDomain (T const &pp) const
 True iff the point pp is in the domain of the basis.
 
void insertKnot (T knot, int mult=1)
 Inserts the knot knot in the underlying knot vector.
 
virtual gsGeometry< T >::uPtr interpolateAtAnchors (gsMatrix< T > const &vals) const
 Applies interpolation of values pts using the anchors as parameter points. May be reimplemented in derived classes with more efficient algorithms. (by default uses interpolateData(pts,vals)
 
memory::unique_ptr< gsGeometry
< T > > 
interpolateData (gsMatrix< T > const &vals, gsMatrix< T > const &pts) const
 Applies interpolation given the parameter values pts and values vals.
 
gsGeometry< T >::uPtr interpolateGrid (gsMatrix< T > const &vals, std::vector< gsMatrix< T > >const &grid) const
 
bool isActive (const index_t i, const gsVector< T > &u) const
 Returns true if there the point u with non-zero value or derivatives when evaluated at the basis function i.
 
bool isClamped () const
 Checks, if both endknots have multiplicity m_p + 1.
 
bool isPeriodic () const
 Tells, whether the basis is periodic.
 
virtual bool isRational () const
 Returns false, since all bases that inherit from gsBasis are not rational.
 
const KnotVectorType & knots (int i=0) const
 Returns the knot vector of the basis.
 
gsMatrix< T > laplacian (const gsMatrix< T > &u) const
 Compute the Laplacian of all nonzero basis functions at points u.
 
gsBasis< T >::domainIter makeDomainIterator () const
 Create a domain iterator for the computational mesh of this basis, that points to the first element of the domain.
 
gsBasis< T >::domainIter makeDomainIterator (const boxSide &s) const
 Create a boundary domain iterator for the computational mesh this basis, that points to the first element on the boundary of the domain.
 
memory::unique_ptr< gsGeometry
< T > > 
makeGeometry (gsMatrix< T > coefs) const
 Create a gsGeometry of proper type for this basis with the given coefficient matrix.
 
virtual memory::unique_ptr
< gsBasis< T > > 
makeNonRational () const
 
void matchWith (const boundaryInterface &, const gsBasis< T > &, gsMatrix< index_t > &, gsMatrix< index_t > &, index_t) const
 Computes the indices of DoFs that match on the interface bi. The interface is assumed to be a common face between this patch and other, with an offset offset. The output is two lists of indices bndThis and bndOther, with indices that match one-to-one on the boundary bi. More...
 
short_t maxDegree () const
 If the basis is of polynomial or piecewise polynomial type, then this function returns the maximum polynomial degree.
 
short_t minDegree () const
 If the basis is of polynomial or piecewise polynomial type, then this function returns the minimum polynomial degree.
 
virtual index_t nPieces () const
 Number of pieces in the domain of definition.
 
int numCrossingFunctions () const
 Returns number of functions crossing the boundary of the knot vector.
 
size_t numElements (boxSide const &s=0) const
 The number of elements on side s.
 
void numElements_cwise (gsVector< unsigned > &result) const
 Returns the number of elements (component wise)
 
unsigned order () const
 Returns the order of the B-spline basis.
 
gsMatrix< T > perCoefs (const gsMatrix< T > &coefs) const
 Helper function for evaluation with periodic basis. More...
 
const gsBasis< T > & piece (const index_t k) const
 Returns the piece(s) of the function(s) at subdomain k.
 
std::ostream & print (std::ostream &os) const
 Prints the object as a string.
 
void reduceContinuity (int const &i=1)
 Reduces spline continuity at interior knots by i.
 
virtual void refine (gsMatrix< T > const &boxes, int refExt=0)
 Refine the basis on the area defined by the matrix boxes. More...
 
void refine_h (short_t const &i=1)
 Uniform h-refinement (placing i new knots inside each knot-span.
 
void refine_k (const TensorSelf_t &other, int const &i=1)
 Increases the degree without adjusting the smoothness at inner knots, except from the knot values in knots (constrained knots of initial geometry) More...
 
void refine_p (short_t const &i=1)
 p-refinement (essentially degree elevation)
 
void refine_withCoefs (gsMatrix< T > &coefs, const std::vector< T > &knots)
 Refine the basis by inserting the given knots and perform knot refinement for the given coefficient matrix.
 
void refine_withTransfer (gsSparseMatrix< T, RowMajor > &transfer, const std::vector< T > &knots)
 Refine the basis by inserting the given knots and produce a sparse matrix which maps coarse coefficient vectors to refined ones.
 
void refineElements (std::vector< index_t > const &elements)
 Refinement function, with different sytax for different basis. More...
 
virtual void refineElements_withCoefs (gsMatrix< T > &coefs, std::vector< index_t > const &boxes)
 Refine basis and geometry coefficients to levels. More...
 
void removeKnot (T knot, int mult=1)
 Removes the knot knot in the underlying knot vector.
 
void reverse ()
 Reverse the basis.
 
void setDegree (short_t const &i)
 Set the degree of the basis (either elevate or reduce) in order to have degree equal to i wrt to each variable.
 
void setDegreePreservingMultiplicity (short_t const &i)
 Set the degree of the basis (either increase or decrecee) in order to have degree equal to i.
 
void setPeriodic (bool flag=true)
 If flag is true, tries to convert the basis to periodic (succeeds only if the knot vector is suitable).
 
index_t size () const
 size More...
 
index_t size (short_t k) const
 The number of basis functions in the direction of the k-th parameter component.
 
template<int s>
void size_cwise (gsVector< index_t, s > &result) const
 The number of basis functions in the direction of the k-th parameter component.
 
virtual const gsBasissource () const
 
virtual gsBasissource ()
 
unsigned stride (short_t dir) const
 Returns the stride for dimension dir.
 
void stride_cwise (gsVector< index_t, d > &result) const
 Returns the strides for all dimensions.
 
gsMatrix< T > support () const
 Returns (a bounding box for) the domain of the whole basis. More...
 
gsMatrix< T > support (const index_t &i) const
 Returns (a bounding box for) the support of the i-th basis function. More...
 
gsMatrix< T > supportInterval (index_t dir) const
 Returns an interval that contains the parameter values in direction dir. More...
 
virtual short_t targetDim () const
 Dimension of the target space. More...
 
gsVector< index_t, d > tensorIndex (const index_t &m) const
 Returns the tensor index of the basis function with global index m.
 
virtual gsBasis::uPtr tensorize (const gsBasis &other) const
 Return a tensor basis of this and other.
 
short_t totalDegree () const
 If the basis is of polynomial or piecewise polynomial type, then this function returns the total polynomial degree.
 
void trimCoefs (gsMatrix< T > &coefs) const
 Helper function for transforming full coefficients to periodic coefficients.
 
int trueSize () const
 Returns the size of the basis ignoring the bureaucratic way of turning the basis into periodic.
 
index_t twin (index_t i) const
 Only meaningfull for periodic basis: For basis members that have a twin, this function returns the other twin index, otherwise it returns the same index as the argument.
 
void uniformCoarsen (int numKnots=1)
 Coarsen the basis uniformly by removing groups of numKnots consecutive knots, each knot removed mul times. More...
 
virtual void uniformCoarsen_withCoefs (gsMatrix< T > &coefs, int numKnots=1)
 Coarsen the basis uniformly. More...
 
void uniformCoarsen_withTransfer (gsSparseMatrix< T, RowMajor > &transfer, int numKnots=1)
 Coarsen the basis uniformly and produce a sparse matrix which maps coarse coefficient vectors to refined ones. More...
 
void uniformRefine (int numKnots=1, int mul=1, int dir=-1)
 Refine the basis uniformly by inserting numKnots new knots with multiplicity mul on each knot span.
 
void uniformRefine_withCoefs (gsMatrix< T > &coefs, int numKnots=1, int mul=1, int dir=-1)
 Refine the basis uniformly. More...
 
void uniformRefine_withTransfer (gsSparseMatrix< T, RowMajor > &transfer, int numKnots=1, int mul=1)
 Refine the basis uniformly. More...
 
virtual const gsMatrix< T > & weights () const
 Only for compatibility reasons, with gsRationalBasis. It returns an empty matrix.
 
virtual gsMatrix< T > & weights ()
 Only for compatibility reasons, with gsRationalBasis. It returns an empty matrix.
 
Evaluation functions
gsVector< index_tnumActive (const gsMatrix< T > &u) const
 Number of active basis functions at an arbitrary parameter value. More...
 
gsMatrix< T > evalSingle (index_t i, const gsMatrix< T > &u) const
 Evaluate a single basis function i at points u.
 
gsMatrix< T > derivSingle (index_t i, const gsMatrix< T > &u) const
 Evaluate a single basis function i derivative at points u.
 
gsMatrix< T > deriv2Single (index_t i, const gsMatrix< T > &u) const
 Evaluate the second derivative of a single basis function i at points u.
 
virtual void numActive_into (const gsMatrix< T > &u, gsVector< index_t > &result) const
 Returns the number of active (nonzero) basis functions at points u in result.
 
virtual void activeCoefs_into (const gsVector< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
 Returns the matrix result of active coefficients at points u, each row being one coefficient. The order of the rows is the same as active_into and eval_into functions. More...
 

Static Public Attributes

static const short_t Dim
 Dimension of the parameter domain.
 

Protected Member Functions

void _convertToPeriodic ()
 Tries to convert the basis into periodic.
 
void _stretchEndKnots ()
 Adjusts endknots so that the knot vector can be made periodic.
 

Protected Attributes

KnotVectorType m_knots
 Knot vector.
 
short_t m_p
 Degree.
 
int m_periodic
 Denotes whether the basis is periodic, ( 0 – non-periodic, >0 – number of ``crossing" functions)
 

Geometry evaluation functions

These functions evaluate not the individual basis functions of the basis, but a geometry object which is represented by a coefficient matrix w.r.t. this basis object. For the format of the coefficient matrix, see gsGeometry.

These functions have default implementations which simply compute the basis function values and perform linear combination, but they may be overridden in derived classes if a higher-performance implementation is possible.

gsMatrix< T > evalFunc (const gsMatrix< T > &u, const gsMatrix< T > &coefs) const
 Evaluate the function described by coefs at points u. More...
 
gsMatrix< T > derivFunc (const gsMatrix< T > &u, const gsMatrix< T > &coefs) const
 Evaluate the derivatives of the function described by coefs at points u. More...
 
virtual void derivFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
 Evaluate the derivatives of the function described by coefs at points u. More...
 
virtual void jacobianFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
 Evaluate the Jacobian of the function described by coefs at points u. Jacobian matrices are stacked in blocks.
 
gsMatrix< T > deriv2Func (const gsMatrix< T > &u, const gsMatrix< T > &coefs) const
 Evaluates the second derivatives of the function described by coefs at points u. More...
 
virtual void deriv2Func_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, gsMatrix< T > &result) const
 Evaluates the second derivatives of the function described by coefs at points u. More...
 
virtual void evalAllDersFunc_into (const gsMatrix< T > &u, const gsMatrix< T > &coefs, const unsigned n, std::vector< gsMatrix< T > > &result, bool sameElement=false) const
 Evaluates all derivatives up to order n of the function described by coefs at points u. More...
 
static void linearCombination_into (const gsMatrix< T > &coefs, const gsMatrix< index_t > &actives, const gsMatrix< T > &values, gsMatrix< T > &result, bool sameElement=false)
 Computes the linear combination coefs * values( actives ) More...
 

Constructor & Destructor Documentation

gsBSplineBasis ( const T  u0,
const T  u1,
const unsigned  interior,
const int  degree,
const unsigned  mult_interior = 1,
const bool  periodic = false 
)
inline

Construct a BSpline basis.

Parameters
u0starting parameter
u1end parameter parameter
interiornumber of interior knots
degreedegree of the spline space
mult_interiormultiplicity at the interior knots
periodicspecifies if basis is periodic or not

Member Function Documentation

gsMatrix<index_t> active ( const gsMatrix< T > &  u) const
inlineinherited

Returns the indices of active (nonzero) functions at points u, as a list of indices.

See Also
active_into()
void active_cwise ( const gsMatrix< T > &  u,
gsVector< index_t, d > &  low,
gsVector< index_t, d > &  upp 
) const
inherited

Returns a box with the coordinate-wise active functions

Parameters
uevaluation points
lowlower left corner of the box
uppupper right corner of the box
void active_into ( const gsMatrix< T > &  u,
gsMatrix< index_t > &  result 
) const
virtualinherited

Returns the indices of active basis functions at points u, as a list of indices, in result. A function is said to be active in a point if this point lies in the closure of the function's support.

Tensor indexing in result
Assume that the parameter domain is three dimensional. Let n1, n2, and n3 denote the number of univariate basis functions in the first, second and third coordinate direction, respectively.
Let the trivariate tensor product basis function B_I be defined by
B_I(x,y,z) = B_i(x) * B_j(y) * B_k(z).
Then, the index I, which is returned in result, is computed as
I = i + j * n1 + k * n1*n2.
Examples:
I <-> (i,j,k)
0 <-> (0,0,0)
1 <-> (1,0,0)
2 <-> (2,0,0)
...
(n1-1) <-> (n1-1,0,0)
n1 <-> (0,1,0)
n1+1 <-> (1,1,0)
n1+2 <-> (2,1,0)
...
n1*n2-1 <-> (n1,n2,0)
n1*n2 <-> (0,0,1)
n1*n2+1 <-> (1,0,1)
...
n1*n2*n3-1 <-> (n1,n2,n3)
Parameters
[in]ugsMatrix containing evaluation points. Each column represents one evaluation point.
[out]resultFor every column i of u, a column containing the active basis functions at evaluation point u.col(i)

Reimplemented from gsTensorBasis< d, T >.

void activeCoefs_into ( const gsVector< T > &  u,
const gsMatrix< T > &  coefs,
gsMatrix< T > &  result 
) const
virtualinherited

Returns the matrix result of active coefficients at points u, each row being one coefficient. The order of the rows is the same as active_into and eval_into functions.

Parameters
[in]ugsVector containing an evaluation point.
[in]coefsgsMatrix is a coefficient matrix with as many rows as the size of the basis
[out]resultFor every column i of u, a column containing the indices of the active basis functions at evaluation point u.col(i).
gsMatrix<T> anchors ( ) const
inlineinherited

Returns the anchor points that represent the members of the basis. There is exactly one anchor point per basis function.

The exact definition of the anchor points depends on the particular basis. For instance, for a Bspline basis these are the Greville abscissae. In general, evaluating a function at the anchor points should provide enough information to interpolate that function using this basis.

const_iterator begin ( ) const
inlineinherited

Get a const-iterator to the beginning of the bases vector

Returns
an iterator to the beginning of the bases vector
iterator begin ( )
inlineinherited

Get an iterator to the beginning of the bases vector

Returns
an iterator to the beginning of the bases vector
gsMatrix< index_t > boundaryOffset ( boxSide const &  s,
index_t  offset 
) const
virtualinherited

Returns the indices of the basis functions that are nonzero at the domain boundary. If an offset is provided (the default is zero), it will return the indizes of the basis functions having this offset to the provided boxSide. Note that the offset cannot be bigger than the size of the basis in the direction orthogonal to boxSide.

Reimplemented from gsBasis< T >.

gsMatrix< index_t > coefSlice ( short_t  dir,
index_t  k 
) const
inherited

Returns all the basis functions with tensor-numbering.

Parameters
kin direction
dir

Detailed explanation:

Tensor-numbering in N-variate tensor-product basis means that each basis function is assigned an identifier (i_0, i_1, ..., i_{N-1}). This function returns indices of basis functions with i_dir = k and the returned indices are numbering of the basis functions in the basis (i.e., 0,1, ..., basis.size() ).

Example:

Bivariate tensor-product basis functions have tensor numbering (a,b). Calling dir=0, k=1 gives all functions with tensor-numbering (1,b). Calling dir=1, k=3 gives all functions with tensor-numbering (a,3).

gsSparseMatrix< T > collocationMatrix ( gsMatrix< T > const &  u) const
inlineinherited

Computes the collocation matrix w.r.t. points u.

The collocation matrix is a sparse matrix with u.cols rows and size() columns. The entry (i,j) is the value of basis function j at evaluation point i.

gsBasis< T >::uPtr componentBasis_withIndices ( boxComponent  b,
gsMatrix< index_t > &  indices,
bool  noBoundary = true 
) const
virtualinherited

Returns the basis that corresponds to the component.

Parameters
bThe component
indicesThe row vector where the indices are stored to
noBoundaryIf true, the transfer matrix does not include parts belonging to lower-order components (i.e., edges without corners or faces without corners and edges)
void compute ( const gsMatrix< T > &  in,
gsFuncData< T > &  out 
) const
virtualinherited

Computes function data.

This function evaluates the functions and their derivatives at the points in and writes them in the corresponding fields of out. Which field to write (and what to compute) is controlled by the out.flags (see also gsFuncData).

The input points in are expected to be compatible with the implementation/representation of the function, i.e. they should be points inside the domain of definitition of the function

Parameters
[in]in
[out]out

Reimplemented in gsGeometry< T >, and gsConstantFunction< T >.

void connectivity ( const gsMatrix< T > &  nodes,
gsMesh< T > &  mesh 
) const
virtualinherited

Returns the connectivity structure of the basis The returned mesh has vertices the rows of matrix nodes

Reimplemented from gsTensorBasis< d, T >.

void connectivityAtAnchors ( gsMesh< T > &  mesh) const
inlinevirtualinherited

Returns the connectivity structure of the basis The returned mesh has the anchor points as vertices

gsMatrix< T > deriv ( const gsMatrix< T > &  u) const
inherited

Evaluate the derivatives,.

See Also
deriv_into()
gsMatrix< T > deriv2 ( const gsMatrix< T > &  u) const
inherited

Evaluates the second derivatives of active (i.e., non-zero) functions at points u.

See documentation for deriv2_into() (the one without input parameter coefs) for details.

See Also
deriv2_into()
Parameters
[in]uEvaluation points in columns.
Returns
For every column of u, a column containing the second derivatives. See documentation for deriv2_into() (the one without input parameter coefs) for details.
void deriv2_into ( const gsMatrix< T > &  u,
gsMatrix< T > &  result 
) const
inlinevirtualinherited

Evaluate the second derivatives of all active basis function at points u.

Input parameter u is a gsMatrix of size d x N, where
d is the dimension of the parameter domain and
N is the number of evaluation points.
Each column of u corresponds to the coordinates of one evaluation point.

result is a gsMatrix of size (K * d) x N, where
K is the number of active basis functions at the evaluation point.
Each column of result corresponds to a column of u. It contains the "pure" and the mixed derivatives for each active basis function, "above" each other.

Example (bivariate): Let \(B_i(x,y)\), d = 2 be bivariate basis functions, and let the functions with indices 3,4,7, and 8 (K = 4) be active at an evaluation point u. Then, the corresponding column of result represents:
\( ( \partial_{xx}\, B_3(u), \partial_{yy}\, B_3(u), \partial_{xy}\, B_3(u), \partial_{xx}\, B_4(u), \partial_{yy}\, B_4(u), \partial_{xy}\, B_4(u), \partial_{xx}\, B_7(u), ... , \partial_{xy}\, B_8(u) )^T \)

Example (trivariate): Let \(B_i(x,y,z)\), d = 3 be trivariate basis functions, and let the functions with indices 3,4,7, and 8 be active at an evaluation point u. Then, the corresponding column of result represents:
\(( \partial_{xx}\, B_3(u), \partial_{yy}\, B_3(u), \partial_{zz}\, B_3(u), \partial_{xy}\, B_3(u), \partial_{xz}\, B_3(u), \partial_{yz}\, B_3(u), \partial_{xx}\, B_4(u), ... , \partial_{yz}\, B_8(u) )^T \)

Parameters
[in]uEvaluation points in columns (see above for format).
[in,out]resultFor every column of u, a column containing the second derivatives as described above.

See also deriv2() (the one without input parameter coefs).

Reimplemented from gsTensorBasis< d, T >.

gsMatrix<T> deriv2Func ( const gsMatrix< T > &  u,
const gsMatrix< T > &  coefs 
) const
inlineinherited

Evaluates the second derivatives of the function described by coefs at points u.

See documentation for deriv2_into() (the one with input parameter coefs) for details.

Parameters
[in]uEvaluation points in columns.
[in]coefsCoefficient matrix describing the geometry in this basis.
Returns
For every column of u, a column containing the second derivatives. See documentation for deriv2_into() (the one with input parameter coefs) for details.
void deriv2Func_into ( const gsMatrix< T > &  u,
const gsMatrix< T > &  coefs,
gsMatrix< T > &  result 
) const
virtualinherited

Evaluates the second derivatives of the function described by coefs at points u.

...i.e., evaluates a linear combination of coefs * (2nd derivatives of basis functions), into result.

Evaluation points u are given as gsMatrix of size d x N, where
d is the dimension of the parameter domain and
N is the number of evaluation points.
Each column of u corresponds to the coordinates of one evaluation point.

The coefficients coefs are given as gsMatrix of size N x n, where
N is the number of points = number of basis functions and
n is the dimension of the physical domain.
Each row of coefs corresponds to the coordinates of one control point.

Let the function \( f: \mathbb R^3 \to \mathbb R^3\) be given by

\[ f = ( f_1, f_2, f_3)^T = \sum_{i=1}^N c_i B_i(x,y,z), \]

where \( B_i(x,y,z)\) are scalar basis functions and \(c_i\) are the corresponding (m-dimensional) control points. Then, for each column in u, the corresponding column in result represents

\[ ( \partial_{xx}\ f_1, \partial_{yy}\ f_1, \partial_{zz}\ f_1, \partial_{xy}\ f_1, \partial_{xz}\ f_1, \partial_{yz}\ f_1, \partial_{xx}\ f_2, \partial_{yy}\ f_2, \ldots , \partial_{xz}\ f_3, \partial_{yz}\ f_3)^T. \]

at the respective evaluation point.

Parameters
[in]uEvaluation points in columns (see above for format).
[in]coefsCoefficient matrix describing the geometry in this basis.
[out]resultFor every column of u, a column containing the second derivatives at the respective point in the format described above.

This function has a default implementation that may be overridden in derived classes for higher performance.
See also deriv2() (the one with input parameter coefs).

void deriv_into ( const gsMatrix< T > &  u,
gsMatrix< T > &  result 
) const
inlinevirtualinherited

Evaluates the first partial derivatives of the nonzero basis function.

Let
d denote the dimension of the parameter domain.
K denote the number of active (i.e., non-zero) basis functions (see active_into()).
N denote the number of evaluation points.
The N evaluation points u are given in a gsMatrix of size d x N. Each column of u represents one evaluation point.

The gsMatrix result contains the computed derivatives in the following form:
Column j of result corresponds to one evaluation point (specified by the j-th column of u). The column contains the gradients of all active functions "above" each other.
For example, for scalar basis functions \(B_i : (x,y,z)-> R\), a column represents
\((dx B_1, dy B_1, dz B_1, dx B_2, dy B_2, dz B_2, ... , dx B_n, dy B_N, dz B_N)^T\),
where the order the basis functions \(B_i\) is as returned by active() and active_into().

Parameters
[in]uEvaluation points given as gsMatrix of size d x N. See above for details.
[in,out]resultgsMatrix of size (K*d) x N. See above for details.
Todo:
Rename to _ grad_into

Reimplemented from gsTensorBasis< d, T >.

gsMatrix<T> derivFunc ( const gsMatrix< T > &  u,
const gsMatrix< T > &  coefs 
) const
inlineinherited

Evaluate the derivatives of the function described by coefs at points u.

Parameters
uevaluation points as N column vectors
coefscoefficient matrix describing the geometry in this basis, n columns
Returns
For every column of u, the result matrix will contain one Jacobian matrix of size d * n, such that the total size of the result is n x (d * n) x N
void derivFunc_into ( const gsMatrix< T > &  u,
const gsMatrix< T > &  coefs,
gsMatrix< T > &  result 
) const
virtualinherited

Evaluate the derivatives of the function described by coefs at points u.

Evaluates a linear combination of coefs*BasisFunctionDerivatives, into result.

This function has a default implementation that may be overridden in derived classes for higher performance.

Let the function \(f: \mathbb R^d \to \mathbb R^m \) be described by the coefficients coefs, i.e.,
each evaluation point is in \(\mathbb R^d\), and
each coefficient is a point in \(\mathbb R^m\).

The N evaluation points u are given in a gsMatrix of size d x N. Each column of u represents one evaluation point.

The K coefficients coefs are given as a gsMatrix of size K x m. Each row of coefs represents one coefficient in \(\mathbb R^m\).

The gsMatrix result contains the following data:
For every column of u, the corresponding column in the matrix result contains the gradients of the m components of the function above each other. Hence, the size of result is (d*m) x N.

Example 1:
Let \(f(s,t)\) be a bivariate scalar function, \(f:\mathbb R^2 \to \mathbb R\) (i.e., d=2, m=1), and let the evaluation point \( u_i\) be represented by the i-th column of u.
Then, result has the form

\[ \left( \begin{array}{cccc} \partial_s f(u_1) & \partial_s f(u_2) & \ldots & \partial_t f(u_{N}) \\ \partial_t f(u_1) & \partial_t f(u_2) & \ldots & \partial_t f(u_{N}) \end{array} \right) \]

Example 2:
Let \(f(s,t) = ( f_1(s,t), f_2(s,t), f_3(s,t) )\) represent a surface in space, \(f:\mathbb R^2 \to \mathbb R^3\) (i.e., d=2, m=3), and let the evaluation point \( u_i\) be represented by the i-th column of u.
Then, result has the form

\[ \left( \begin{array}{ccccccc} \partial_s f_1(u_1) & \partial_s f_1(u_2) & \ldots & \partial_s f_1(u_N) \\ \partial_t f_1(u_1) & \partial_t f_1(u_2) & \ldots & \partial_t f_1(u_N) \\ \partial_s f_2(u_1) & \partial_s f_2(u_2) & \ldots & \partial_s f_2(u_N) \\ \partial_t f_2(u_1) & \partial_t f_2(u_2) & \ldots & \partial_t f_2(u_N) \\ \partial_s f_3(u_1) & \partial_s f_3(u_2) & \ldots & \partial_s f_3(u_N) \\ \partial_t f_3(u_1) & \partial_t f_3(u_2) & \ldots & \partial_t f_3(u_N) \\ \end{array} \right) \]

Parameters
[in]uEvaluation points as d x N-matrix.
[in]coefsCoefficient matrix describing the geometry in this basis as K x m-matrix.
K should equal the size() of the basis, i.e., the number basis functions.
[in,out]resultgsMatrix of size d*m x N, see above for format.

where
d is the dimension of the parameter domain
m is the dimension of the physical domain
N is the number of evaluation points
K is the number of coefficients

void derivSingle_into ( index_t  i,
const gsMatrix< T > &  u,
gsMatrix< T > &  result 
) const
inlinevirtualinherited

Evaluates the (partial) derivatives of the i-th basis function at points u into result.

See deriv_into() for detailed documentation.

Todo:
rename grad_into

Reimplemented from gsTensorBasis< d, T >.

gsDomain<T>* domain ( ) const
inlinevirtualinherited

Return the gsDomain which represents the parameter domain of this basis. Currently unused.

Reimplemented from gsBasis< T >.

short_t domainDim ( ) const
inlinevirtualinherited

Dimension of the (source) domain.

Returns
For \(f:\mathbb{R}^n\rightarrow\mathbb{R}^m\), returns \(n\).

Implements gsFunctionSet< T >.

const_iterator end ( ) const
inlineinherited

Get a const-iterator to the end of the bases vector

Returns
an iterator to the end of the bases vector
iterator end ( )
inlineinherited

Get an iterator to the end of the bases vector

Returns
an iterator to the end of the bases vector
gsMatrix< T > eval ( const gsMatrix< T > &  u) const
inherited

Evaluate the function,.

See Also
eval_into()
void eval_into ( const gsMatrix< T > &  u,
gsMatrix< T > &  result 
) const
virtualinherited

Evaluates nonzero basis functions at point u into result.

Let...
d denote the dimension of the parameter domain.
K denote the number of active (i.e., non-zero) basis functions (see active_into()). N denote the number of evaluation points.
The n evaluation points u are given in a gsMatrix of size d x N. Each column of u represents one evaluation point.

The gsMatrix result contains the computed function values in the following form:
Column j of result corresponds to one evaluation point (specified by the j-th column of u). The column contains the values of all active functions "above" each other.
For example, for scalar basis functions Bi : (x,y,z)-> R, a column represents
(B1, B2, ... , BN)^T,
where the order the basis functions Bi is as returned by active() and active_into().

Parameters
[in]uEvaluation points given as gsMatrix of size d x N. See above for details.
[in,out]resultgsMatrix of size K x N. See above for details.

Reimplemented from gsTensorBasis< d, T >.

std::vector< gsMatrix< T > > evalAllDers ( const gsMatrix< T > &  u,
int  n,
bool  sameElement = false 
) const
inherited

Evaluate all derivatives upto order n,.

See Also
evalAllDers_into
void evalAllDers_into ( const gsMatrix< T > &  u,
int  n,
std::vector< gsMatrix< T > > &  result,
bool  sameElement = false 
) const
virtualinherited

Evaluate the nonzero functions and their derivatives up to order n at points u into result.

The derivatives (the 0-th derivative is the function value) are stored in a result. result is a std::vector, where result[i] is a gsMatrix which contains the i-th derivatives.

The entries in result[0], result[1], and result[2] are ordered as in eval_into(), deriv_into(), and deriv2_into(), respectively. For i > 2, the derivatives are stored in lexicographical order, e.g. for order i = 3 and dimension 2 the derivatives are stored as follows: \( \partial_{xxx}, \, \partial_{xxy}, \, \partial_{xyy}, \, \partial_{yyy}.\, \)

Parameters
[in]uEvaluation points, each column corresponds to one evaluation point.
[in]nAll derivatives up to order n are computed and stored in result.
[in,out]resultSee above for format.

Reimplemented from gsTensorBasis< d, T >.

void evalAllDersFunc_into ( const gsMatrix< T > &  u,
const gsMatrix< T > &  coefs,
const unsigned  n,
std::vector< gsMatrix< T > > &  result,
bool  sameElement = false 
) const
virtualinherited

Evaluates all derivatives up to order n of the function described by coefs at points u.

Evaluation points u are given as gsMatrix of size d x N, where
d is the dimension of the parameter domain and
N is the number of evaluation points.
Each column of u corresponds to the coordinates of one evaluation point.

The coefficients coefs are given as gsMatrix of size K x n, where
K is the number of (active) basis functions (=size()) and
n is the dimension of the physical domain.
Each row of coefs corresponds to the coordinates of one control point.

result is a std::vector, where the entry result[i] contains the gsMatrix corresponding to the i-th derivatives. The format of the respective entry is as in
evalFunc_into()
derivFunc_into()
deriv2Func_into()

Todo:
finish documentation
Parameters
[in]u
[in]coefs
[in]n
[out]result
gsMatrix<T> evalFunc ( const gsMatrix< T > &  u,
const gsMatrix< T > &  coefs 
) const
inlineinherited

Evaluate the function described by coefs at points u.

This function has a default implementation that may be overridden in derived classes for higher performance.

Parameters
uevaluation points as m column vectors
coefscoefficient matrix describing the geometry in this basis, n columns
Returns
a matrix of size n x m with one function value as a column vector per evaluation point
void evalFunc_into ( const gsMatrix< T > &  u,
const gsMatrix< T > &  coefs,
gsMatrix< T > &  result 
) const
inlinevirtualinherited

Evaluate the function described by coefs at points u, i.e., evaluates a linear combination of coefs x BasisFunctions, into result.

This function has a default implementation that may be overridden in derived classes for higher performance.

Parameters
uevaluation points as N column vectors
coefscoefficient matrix describing the geometry in this basis, n columns
[out]resulta matrix of size n x N with one function value as a column vector per evaluation point

Reimplemented from gsBasis< T >.

gsBasisFun< T > function ( index_t  i) const
inherited

Returns the i-th basis function as a gsFunction.

Note that the gsBasisFun object only holds a reference to the current basis, so it is invalidated when the basis is destroyed.

unsigned index ( unsigned  i,
unsigned  j,
unsigned  k = 0 
) const
inlineinherited

Returns the global index of the basis function created by components of indices i,j,k (for 2d or 3d only)

index_t index ( gsVector< index_t, d > const &  v) const
inlineinherited

Returns the global index of the basis function created by components of indices given in the vector v

gsGeometry< T >::uPtr interpolateGrid ( gsMatrix< T > const &  vals,
std::vector< gsMatrix< T > >const &  grid 
) const
inherited

Interpolates values on a tensor-grid of points, given in tensor form (d coordinate-wise vectors). Samples vals should be ordered as the tensor-basis coefficients

void linearCombination_into ( const gsMatrix< T > &  coefs,
const gsMatrix< index_t > &  actives,
const gsMatrix< T > &  values,
gsMatrix< T > &  result,
bool  sameElement = false 
)
staticinherited

Computes the linear combination coefs * values( actives )

Todo:
documentation
Parameters
[in]coefsgsMatrix of size K x m, where K should equal size() of the basis (i.e., the number of basis functions).
[in]activesgsMatrix of size numAct x numPts
[in]valuesgsMatrix of size stride*numAct x numPts
[out]resultgsMatrix of size stride x numPts
virtual memory::unique_ptr<gsBasis<T> > makeNonRational ( ) const
inlinevirtualinherited

Clone the source of this basis in case of rational basis, same as clone() otherwise

Reimplemented in gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, and gsRationalBasis< gsBSplineBasis< T > >.

void matchWith ( const boundaryInterface bi,
const gsBasis< T > &  other,
gsMatrix< index_t > &  bndThis,
gsMatrix< index_t > &  bndOther,
index_t  offset 
) const
inlinevirtualinherited

Computes the indices of DoFs that match on the interface bi. The interface is assumed to be a common face between this patch and other, with an offset offset. The output is two lists of indices bndThis and bndOther, with indices that match one-to-one on the boundary bi.

NOTE: bndThis will have offset but bndOther will NOT have an offset (hence offset 0)

Reimplemented from gsBasis< T >.

gsVector<index_t> numActive ( const gsMatrix< T > &  u) const
inlineinherited

Number of active basis functions at an arbitrary parameter value.

Usually, this is used for getting the active functions on one element, assuming that this number doesn't change for different parameters inside the element.

gsMatrix<T> perCoefs ( const gsMatrix< T > &  coefs) const
inlineinherited

Helper function for evaluation with periodic basis.

Parameters
coefscoefficients (control points, one per row) before converting the basis into periodic.
Returns
copy of coefs with the first m_periodic rows copied to the last m_periodic rows.
void refine ( gsMatrix< T > const &  boxes,
int  refExt = 0 
)
virtualinherited

Refine the basis on the area defined by the matrix boxes.

boxes is a d x n-matrix (n even), where d is the dimension of the parameter domain.
n must be even, and every 2 successive columns in the matrix define a box in the parameter domain (the first column represents the coordinates of the lower corner, the second column the coordinates of the upper corner).

Example: The input of the matrix

\[ \left[\begin{array}{cccc} 0 & 0.2 & 0.8 & 1 \\ 0.4 & 0.6 & 0.2 & 0.4 \end{array} \right] \]

results in refinement of the two boxes \([0,0.2]\times[0.4,0.6]\) and \([0.8,1]\times[0.2,0.4]\).

Parameters
[in]boxesgsMatrix of size d x n, see above for description of size and meaning.
[in]refExtExtension to be applied to the refinement boxes

Reimplemented in gsHTensorBasis< d, T >, gsTensorBSplineBasis< d, T >, and gsTensorBSplineBasis< domainDim+1, T >.

void refine_k ( const TensorSelf_t other,
int const &  i = 1 
)
inlineinherited

Increases the degree without adjusting the smoothness at inner knots, except from the knot values in knots (constrained knots of initial geometry)

This type of refinement is known as k-refinement. Note that this type of refinement is ment to be performed after one (or more) h-refinement steps the parent mesh (other), otherwise this function is equivalent to p-refinement of other.

Parameters
otherparent/reference mesh determining the smoothness at the inner knots.
inumber of k-refinement steps to perform
Remarks
Not tested yet!
void refineElements ( std::vector< index_t > const &  boxes)
inlinevirtualinherited

Refinement function, with different sytax for different basis.

See documentation of
gsTensorBasis::refineElements()
gsHTensorBasis::refineElements()

Reimplemented from gsBasis< T >.

void refineElements_withCoefs ( gsMatrix< T > &  coefs,
std::vector< index_t > const &  boxes 
)
virtualinherited

Refine basis and geometry coefficients to levels.

Refines the basis as well as the coefficients. The refinement and the format of the input depend on the implementation of refineElements().

Reimplemented in gsHTensorBasis< d, T >, and gsRationalBasis< SrcT >.

index_t size ( ) const
inlinevirtualinherited

size

Warning
gsFunction and gsGeometry have size() == 1. This should not be confused with the size eg. of gsGeometry::basis(), which is the number of basis functions in the basis
Returns
the size of the function set: the total number of functions

Reimplemented from gsFunctionSet< T >.

virtual const gsBasis& source ( ) const
inlinevirtualinherited

Applicable for rational bases: returns the underlying "source" (non-rational) basis

Reimplemented in gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, and gsRationalBasis< gsBSplineBasis< T > >.

virtual gsBasis& source ( )
inlinevirtualinherited

Applicable for rational bases: returns the underlying "source" (non-rational) basis

Reimplemented in gsRationalBasis< SrcT >, gsRationalBasis< gsBSplineTraits< d, T >::Basis >, and gsRationalBasis< gsBSplineBasis< T > >.

gsMatrix< T > support ( ) const
virtualinherited

Returns (a bounding box for) the domain of the whole basis.

Returns a dx2 matrix, containing the two diagonally extreme corners of a hypercube.

Reimplemented from gsBasis< T >.

gsMatrix< T > support ( const index_t i) const
virtualinherited

Returns (a bounding box for) the support of the i-th basis function.

Returns a dx2 matrix, containing the two diagonally extreme corners of a hypercube.

Reimplemented from gsBasis< T >.

gsMatrix< T > supportInterval ( index_t  dir) const
inherited

Returns an interval that contains the parameter values in direction dir.

Returns a 1x2 matrix, containing the two endpoints of the interval.

void uniformCoarsen ( int  numKnots = 1)
inlinevirtualinherited

Coarsen the basis uniformly by removing groups of numKnots consecutive knots, each knot removed mul times.

This function is the oposite of gsBasis::uniformRefine

The execution of

basis->uniformRefine (nKnots, mul)
basis->uniformCoarsen(nKnots);

results in no overall change in "basis". However,

basis->uniformCoarsen(nKnots);
basis->uniformRefine (nKnots, mul)

is not guaranteed to keep "basis" unchanged.

See Also
gsBasis::uniformRefine

Reimplemented from gsTensorBasis< d, T >.

void uniformCoarsen_withCoefs ( gsMatrix< T > &  coefs,
int  numKnots = 1 
)
virtualinherited

Coarsen the basis uniformly.

The function simultainously updates the vector coefs, representing a function in the bases, such that its new version represents the same function.

This function is equivalent to

gsSparseMatrix<T,RowMajor> transfer;
basis->uniformCoarsen_withTransfer(transfer, numKnots);
coefs = transfer * coefs;
See Also
gsBasis::uniformRefine

Reimplemented in gsHTensorBasis< d, T >.

void uniformCoarsen_withTransfer ( gsSparseMatrix< T, RowMajor > &  transfer,
int  numKnots = 1 
)
virtualinherited

Coarsen the basis uniformly and produce a sparse matrix which maps coarse coefficient vectors to refined ones.

The function writes a sparse matrix into the variable transfer that indicates how the functions on the coarse grid are represented as linear combinations as fine grid functions

See Also
gsBasis::uniformCoarsen

Reimplemented from gsBasis< T >.

void uniformRefine_withCoefs ( gsMatrix< T > &  coefs,
int  numKnots = 1,
int  mul = 1,
int  dir = -1 
)
virtualinherited

Refine the basis uniformly.

The function simultainously updates the vector coefs, representing a function in the bases, such that its new version represents the same function.

This function is equivalent to

gsSparseMatrix<T,RowMajor> transfer;
basis->uniformRefine_withTransfer(transfer, numKnots, mul);
coefs = transfer * coefs;
See Also
gsBasis::uniformRefine

Reimplemented from gsBasis< T >.

void uniformRefine_withTransfer ( gsSparseMatrix< T, RowMajor > &  transfer,
int  numKnots = 1,
int  mul = 1 
)
virtualinherited

Refine the basis uniformly.

The function writes a sparse matrix into the variable transfer that indicates how the functions on the coarse grid are represented as linear combinations as fine grid functions

See Also
gsBasis::uniformRefine

Reimplemented from gsBasis< T >.